is a smooth function.

A real hypersurface is said to be a Hopf hypersurface if

is principal.

The Jacobi operator field with respect to X on M is de-

fined by

,RXX

=X

, where R is the Riemmanian curva-

ture of M. For

the Jacobi operator is called

structure Jacobi operator and is denoted by

=,lR

.

It has a fundamental role in almost contact manifolds.

Many differential geometers have studied real hypersur-

faces in terms of the structure Jacobi operator.

The Lie derivative of the structure Jacobi operator

with respect to

was investigated by Perez, Santos,

Suh (see [7]). More precisely, they classified real hyper-

surfaces in n

P

3

=0l

2

(n), whose structure Jacobi op-

erator satisfies the condition:

L. Ivey and Ryan

(see [8]) classified real hypersurfaces satisfying the same

condition in

P

2

and

H

.

The study of real hypersurfaces whose structure Jacobi

operator is parallel is a problem of great importance. In

[9] the nonexistence of real hypersurfaces in nonflat

C

opyright © 2012 SciRes. APM

K. PANAGIOTIDOU ET AL.

2

,= ,,

,= ,

complex space form with parallel structure Jacobi opera-

tor () was proved. In [10] a weaker condition

(-parallelness), X for any vector field X or-

thogonal to

g

XY gXYXY

gX YgXY

=,

=,

X

X

AX

YYAXgAXY

=0l

=0l

, was studied and it was proved the non-

existence of such hypersurfaces in case of n

P

3

n

(n).

The parallelness of structure Jacobi operator in combina-

tion with other conditions was another problem that was

studied by many others such as Ki, Kim, Perez, Santos,

Suh (see [11,12]).

Recently Perez-Santos (see [1]) studied real hypersur-

faces in

P

for , whose structure Jacobi opera-

tor satisfies the relation:

3n

=.ll

L

P H

P

H

(1)

In the present paper we go on studying the same prob-

lem for 2

and 2. We prove the following theo-

rem:

Main Theorem Let M be a real hypersurface in 2

or 2, whose structure Jacobi operator satisfies rela-

tion (1). Then M is locally congruent to: a geodesic

sphere of radius r, where π

<2

r0< with π

4

r, or to

a tube of radius π

=4

r over a holomorphic curve in

2 and to a horosphere, a geodesic sphere or a tube

over

P

1

H

A

in 2i or to a Hopf hypersurface in 2

with

HH

=0

.

2. Preliminaries

Let M be a connected real hypersurface immersed in a

nonflat complex space form

,

n

M

cG 0c

=

, (), with

almost complex structure J of constant holomorphic sec-

tional curvature c. Let N be a unit normal vector field on

M and

J

N

. For a vector field X tangent to M we

can write

=

J

XX

XN

, where φX and

X

N

are the tangential and the normal component of JX re-

spectively. The Riemannian connection

in

n

M

c

and in M are related for any vector fields X, Y on M:

=,

YY

X

XgAYXN

=

XNAX

where g is the Riemannian metric on M induced from G

of

n

M

c and A is the shape operator of M in

n

M

c

(,,)

.

M has an almost contact metric structure

in-

duced from J on

n

M

c where

is a (1,1) tensor field

and

is a 1-form on M such that

,= ,,

g

XY

G JXY

= ,

=,

X

gX

GJXN

=0,X

(see [13]). Then we have

2=,

=0, =1

XX X

(2)

(3)

(4)

Since the ambient space is of constant holomorphic

sectional curvature c, the equations of Gauss and Codazzi

for any vector fields X, Y, Z on M are respectively given

by

,= ,,,

4

,2,

,,

c

RXYZgYZXgXZY gYZX

gXZYgXYZ

gAYZAXgAXZAY

(5)

=4

2,

XY

c

A

YAX XYYX

gXY

PM

(6)

where R denotes the Riemannian curvature tensor on M.

For every point

, the tangent space can

be decomposed as following:

P

TM

=

P

T Mspanker

where

=:=0kerXTMX

P. Due to the

above decomposition, the vector field

A

is decom-

posed as follows:

A

U=

(7)

where =

and

1

=U ker

0

, pro-

vided that

.

All manifolds are assumed connected and all mani-

folds, vector fields etc are assumed smooth (C

).

3. Auxiliary Relations

2

P

2

or Suppose now that the ambient space is

H

,

(i.e.

2c0c,

M

), then we consider V be the open

subset of points P

M, such that there exists a neighbor-

hood of every P, where =0

and the open subset

of points Q of M such that there exists a neighborhood of

every Q, where

0a

. Since,

is a smooth function

on M, then is an open and dense subset of M. V

Proposition 3.1 Let M be a real hypersurface in

2

M

c, equipped with structure Jacobi operator satis-

fying (1). Then,

is principal on V.

A

Proof: The relation (7) on V, takes the form

U

X

UYZ

. From (5) for

,

we obtain:

.

4

c

lU U (8)

Copyright © 2012 SciRes. APM

K. PANAGIOTIDOU ET AL. 3

Due to the definition of Lie derivative, the relation (1)

for X

yields: l

0

0

. The latter, because of the

first relation of (4) and (8) implies

A

, hence

0

. Therefore,

is principal on V. □

On if

=0

, then

is principal. In what fol-

lows we work on W (W), which is the open subset

of points such that

0

Q

.

Lemma 3.2 Let M be a real hypersurface in

2

M

c.

Then the following relations hold on W:

2

,

44

cc

A

UU

AUU

(9)

2

,

44

UU

cc

UU

,

U

(10)

12

,,

4

UU

c

UUU U

3

UU

(11)

2

23

,

UU

14

,

U

U

c

UU

UU

3

,,

,,UU

(12)

where are smooth functions on M.

12

Proof: If

=,=,

is an orthonormal basis, then

because of (7) we have:

A

UU UAUUU

(13)

where

,

and

are smooth functions on M.

The first relation of (4), because of (13) yields:

=,

=.

U

U

= ,

U

UU UU

(14)

The relation (5), using (13) can be written:

2

4

c

lU

lU U

,

.

4

U U

cU

.

lX X

l

(15)

By the definition of Lie derivative, the relation (1)

takes the form:

The latter for X

,U

0lU implies lU = 0,

and then from (15) we obtain:

2

44

cc

0, ,.

,

XX

(16)

The relations (13) and (14), because of (16) imply (9)

and (10) respectively.

From the well known relation:

,,

X

gYZgYZ

,,

gY Z

for

X

YZ

,,UU

, using (16) we obtain (11) and

(12), where 1

, 2

and 3

are smooth functions on M.

□

,,UU

The Codazzi equation for X, Y , because

of Lemma 3.2 yields:

2

2

41,Uc

(17)

22

3

1,

44

cc

(18)

2

2

4,Uc

(19)

2

2

4,

c

(20)

3

3,

4

c

U

(21)

2

2

1,

24

cc

U

(22)

22

13.

44

cc

U

(23)

The Riemannian curvature on M satisfies (5) and on

the other hand is given by the relation

,RXYZZZ Z

,,UU

,

XY YXXY . From these two

relations and because of (16) for X, Y we

obtain:

2

3

21

2

22

3

12

242

c

cc

UU

c

(24)

2

312 3

4

c

U

(25)

3213 3

42

cc

U

34

(26)

Relation (23), because of (18), (21) and (22), yields:

(27)

and so relation (18) becomes:

2

2

14.

44

cc

(28)

Differentiating the relations (27) and (28) with respect

to U and

respectively and substituting in (25) and

due to (19), (20) and (27) we obtain:

22

224 0c

. (29)

Copyright © 2012 SciRes. APM

K. PANAGIOTIDOU ET AL.

4

Owing to (29), we consider , (W) the open

subset of points , where in a neighbor-

hood of every Q.

1

W1W

20

QW

Lemma 3.3 Let M be a real hypersurface in

2

M

c

1

W

2

c

,

equipped with Jacobi operator satisfying (1). Then

is empty.

Proof: Due to (29) we obtain: on 1

W.

Differentiation of the last relation along

2

24

and taking

into account (19), (20) and yields: ,

which is a contradiction. Therefore, is empty. □

2

24

2

c

=0c

2=0

0.

1

On W, because of Lemma 3.3, we have , hence

the relations (17), (19) and (20) become:

W

UU

Using the last relations and Lemma 3.2 we obtain:

,0,U

UU

.cU

22

,

1416

4

U

UU

0cU

WWQW

Combining the last two relations we have:

22

416

. (30)

Let 2, (2) be the set of points W

, for

which there exists a neighborhood of every Q such that

. So from (30) we have: 16 .

Differentiating the last relation with respect to

U

0 22

4c

U

and

taking into account (21), (22), (27) and (28), we have:

, which is impossible. So 2 is empty. Hence,

on W we have . Then, relations (21) and

(28), because of (27) imply respectively:

20

W

U

0

2

4c

2

and

2

5

1

. Relation (26), because of

0U

,

2

4c

and (27) yields: 1

2

. Substitution of 1

in 1

2

52

yields: 322

. Differentiation of

the last one along U

and taking into account (22) leads

to: 0

, which is a contradiction. So we obtain the

following proposition:

Proposition 3.4 Let M be a real hypersurface in

2

M

c, equipped with Jacobi operator satisfying (1).

Then M is a Hopf hypersurface.

4. Proof of Main Theorem

Since M is a Hopf hypersurface, we can write

A

ZZ

and

A

ZZ

with

,,

Z

Z

, being a local or-

thonormal basis and the following relation holds:

24

c

(Corollary 2.3 [14]). Furthermore,

due to Theorem 2.1 [14], we have that

is constant.

From (5), due to

A

ZZ

and

A

ZZ

we

get:

,.

44

cc

lZZl ZZ

(31)

The relation (1), because of (31) implies:

0

, for

X

Z

and , for

0

X

Z

. Combining the last two relations leads to:

20

2

P

, M is locally

0

. If

, in case of

congruent to a tube of radius π

4

2

over a holomorphic

P

curve in

0A

due to [15] and to a Hopf hypersurface

with 2

H

. If

in 0

, the last relation im-

plies:

and we obtain:

0, ,

A

AX XTM

which because of Theorem A completes the proof of

Main Theorem.

5. Acknowledgements

The first author is granted by Foundation Alexandros S.

Onasis. Grant Nr: G ZF 044/2009-2010. The authors

would like to thank Professors Juan de Dios Perez and

Patrick J. Ryan. Their answers improved the proof of

main theorem. Finally, the authors express their grati-

tude to the referee who has contributed to improve the

paper.

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